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Understanding an integral rule.

  1. Jul 22, 2015 #1
    Hi I'm tying to understand notations in Integration, and would really appreciate some help making sure that my understanding is right.

    My books writes

    Let u and v be functions of x whose domains are an open interval I, and suppose du and dv exist for every x in I.

    Then it defines

    1) ∫(du)= u + C
    2) ∫(c*du) = c*∫(du)

    3) ∫(cos(u)du = sin u +C

    Now i do understand the first 2, but I want to make sure i understand the 3rd rule.

    If u is a function of x with the equation u(x)=x^2

    Then the derivative
    du/dx= 2x

    The differential

    Now if it's true that du=u'(x)dx
    Then it does make sense that ∫du =u+C because ∫du=∫u'(x)*dx and the integral of the derivative if u is u.

    But if u=x^2

    Then ∫(cos(x^2)*du = ∫(cos(x^2)*(2x)dx and this is = sin(x^2) + C as the statement above says.
    because the derivative of sin(x^2) = cos(x^2)*(2x).

    Is this the right interpretation?
    Last edited: Jul 22, 2015
  2. jcsd
  3. Jul 22, 2015 #2
    why is ∫(cos(x^2)*(2x)dx not equal to sin(x^2) + C? you said in the line below it that
    which is exactly what appears under the integral sign.
  4. Jul 22, 2015 #3
    Oh forgive me, that NOT was a BIG mistake ;)
    You are right, it is e equal to it.
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